Código:

<?php
$comments_count = wp_count_comments($post->ID);
echo " $comments_count->approved ";
?>

Total de comentarios en el website:

<?php
$comments_count = wp_count_comments();
echo "Comments for site <br />";
echo "Comments in moderation: " . $comments_count->moderated . "<br />";
echo "Comments approved: " . $comments_count->approved . "<br />";
echo "Comments in Spam: " . $comments_count->spam . "<br />";
echo "Comments in Trash: " . $comments_count->trash . "<br />";
echo "Total Comments: " . $comments_count->total_comments . "<br />";
?>

Numero de comentarios por post:

<?php
$comments_count = wp_count_comments( 2492 );
echo "Comments for post 2492 <br />";
echo "Comments in moderation: " . $comments_count->moderated . "<br />";
echo "Comments approved: " . $comments_count->approved . "<br />";
echo "Comments in Spam: " . $comments_count->spam . "<br />";
echo "Comments in Trash: " . $comments_count->trash . "<br />";
echo "Total Comments: " . $comments_count->total_comments . "<br />";
?>

Referencia:

http://codex.wordpress.org/Function_Reference/wp_count_comments